# q-series : their development and application in analysis, number theory, combinatorics, physics, and computer algebra

@inproceedings{Andrews1986qseriesT, title={q-series : their development and application in analysis, number theory, combinatorics, physics, and computer algebra}, author={George E. Andrews}, year={1986} }

Found opportunities Classical special functions and L. J. Rogers W. N. Bailey's extension of Roger's work Constant terms Integrals Partitions and $q$-series Partitions and constant terms The hard hexagon model Ramanujan Computer algebra Appendix A. W. Gosper's Proof that $\lim_{q\rightarrow 1^-}\Gamma_q(x)=\Gamma (x)$ Appendix B. Roger's symmetric expansion of $\psi (\lambda, \mu,\nu, q, \theta)$ Appendix C. Ismail's proof of the $_1\psi_1$-summation and Jocobi's triple product identity… Expand

#### 500 Citations

qFunctions - A Mathematica package for q-series and partition theory applications

- Computer Science, Mathematics
- J. Symb. Comput.
- 2021

The qFunctions Mathematica package can symbolically handle formal manipulations onq-differential, $q-shift equations and recurrences, such as switching between these forms, finding the greatest common divisor of recurrence, and formal substitutions. Expand

Eulerian series, zeta functions and the arithmetic of partitions

- Mathematics, Physics
- 2020

In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory,… Expand

Constructive extensions of three summation formulas for q-series and their applications to Bailey pairs

- Mathematics
- 2017

This paper is devoted to three constructive extensions associated with the Pfaff–Saalschütz $${}_3\phi _2$$3ϕ2, Watson’s $${}_6\phi _5$$6ϕ5, and Jackson’s $${}_8\phi _7$$8ϕ7 summation formulas in… Expand

Renormalization and quantum modular forms, part II: Mock theta functions

- Mathematics
- 2013

Sander Zwegers showed that Ramanujan's mock theta functions are $q$-hypergeometric series, whose $q$-expansion coefficients are half of the Fourier coefficients of a non-holomorphic modular form.… Expand

Elliptic genera and q-series development in analysis, string theory, and N=2 superconformal field theory

- Physics
- 2019

In this article we examine the Ruelle type spectral functions $\cR(s)$,which define an overall description of the content of the work. We investigate the Gopakumar-Vafa reformulation of the string… Expand

A_2 Macdonald polynomials: a separation of variables

- Mathematics
- 1995

In this paper we construct a discrete linear operator $K$ which transforms $A_2$ Macdonald polynomials into the product of two basic $3\phi_2$ hypergeometric series with known arguments. The action… Expand

Conjugate Bailey pairs

- Mathematics
- 1999

In this paper it is shown that the one-dimensional configuration sums of the
solvable lattice models of Andrews, Baxter and Forrester and the string functions
associated with admissible… Expand

Further results on Andrews--Yee's two identities for mock theta functions $\omega(z;q)$ and $v(z;q)$

- Mathematics
- 2018

In this paper, by the method of comparing coefficients and the inverse technique, we establish the corresponding variate forms of two identities of Andrews and Yee for mock theta functions, as well… Expand

A New $A_n$ Extension of Ramanujan's ${}_1\psi_1$ Summation with Applications to Multilateral An Series

- Mathematics
- 2000

In this article we derive some identities for multilateral basic hypergeometric series associated to the root system An. First, we apply Ismail's [15] argument to an An q-binomial theorem of Milne… Expand

On some special families of $q$-hypergeometric Maass forms

- Mathematics
- 2016

Using special polynomials introduced by Hikami and the second author in their study of torus knots, we construct classes of $q$-hypergeometric series lying in the Habiro ring. These give rise to new… Expand

#### References

SHOWING 1-10 OF 13 REFERENCES

Projection Formulas, a Reproducing Kernel and a Generating Function for q-Wilson Polynomials

- Mathematics
- 1985

A projection formula for the q-Wilson polynomials $p_n (x;a,b,c,d)$ is obtained which is then used to construct a reproducing kernel. Using Askey and Wilson’s q-analogue of the beta integral an… Expand

Asymptotic Formulas for Zero-Balanced Hypergeometric Series

- Mathematics
- 1984

A hypergeometric series is called s-balanced if the sum of denominator parameters minus the sum of numerator parameters is s. A nonterminating s-balanced hypergeometric series converges at $x = 1$ if… Expand

[ 1 ] Beitrage zur theorie der durch die heinesche reihe ; 1 + ( ( 1 — q a ) ( l — qP ) / ( \ — ^ ) ( 1 — q y ) ) x + • • * darstellbaren functionen

- Symbolic mathematical computation , Encyclopedi a o f Compute r Science and Technology
- 1980

4t h ed

- , Oxfor d Universit y Press, London and New York,
- 1960

[ 1 ] Bemerkninger om et multipelt integral ,

- Norsk e Mat . Tidsskr .
- 1944

[ 1 ] Treue Darstellung Liescher Ringe

- J . Rein e Angew . Math .
- 1937